Freddy Saia

Research

My research interests lie in number theory and arithmetic geometry. In particular, I am interested in both theoretical and computational aspects of the arithmetic of elliptic curves and higher-dimensional abelian varieties, modular and Shimura curves, Galois representations, and low-degree points on varieties.

I am also interested in involving students in research. In the 2024-2025 academic year, I mentored UIC undergrad Raghav Bhutani’s project titled “Replacement dynamics of binary quadratic forms,” supported by the LASURI program.

Preprints

• Shimura curve Atkin–Lehner quotients of genus at most two (computations), (arXiv)
  with Oana Padurariu

• Explicit surjectivity of Galois representations of products of elliptic curves over function fields (arXiv)
  with Alina Cojocaru

• Point counts, automorphisms, and gonalities of Shimura curves (computations), (arXiv)
  with Pietro Mercuri, Oana Padurariu, and Claudio Stirpe

• Replacement dynamics of binary quadratic forms (arXiv)
  with Raghav Bhutani

• CM elliptic curves: volcanoes, reality and applications, part II (preprint)
  with Pete L. Clark

Publications

• Counting points on Atkin–Lehner quotients of some genus \(0\) Shimura curves (arXiv), (pub)
  with Tyler Genao, Tristan Phillips, Tim Santens, and John Yin
  Mathematical Proceedings of the Cambridge Philisophical Society, to appear

• Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level (computations), (arXiv), (pub)
  with Oana Padurariu
  Research in Number Theory 11, 2 (2025)

• CM points on Shimura curves via QM-equivariant isogeny volcanoes (computations), (arXiv), (pub)
  Pacific Journal of Mathematics, Vol. 332 (2024), no. 2, 321–384

• The least degree of a CM point on a modular curve (computations), (pdf), (pub)
  with Pete L. Clark, Tyler Genao, and Paul Pollack
  Journal of the London Mathematical Society (2) 105 (2022), no. 2, 825–883

• Chevalley-Warning at the boundary (pdf), (pub)
  with Pete L. Clark and Tyler Genao
  Expositiones Mathematicae 39 (2021), no. 4, 604–623

• Classically integral quadratic forms excepting at most two values (computations), (arXiv), (pub)
  with Madeleine Barowsky, Will Damron, Andres Mejia, Nolan Schock, and Kate Thompson
  Proceedings of the AMS 146 (2018), no. 9, 3661–3677

Conferences and Workshops

Invited and Contributed Talks

• Effective surjectivity of Galois representations of products of elliptic curves over function fields
  AMS Sectional Special Session on Counting and Asymptotics in Number Theory, Hartford, CT – April 2025
  Number Theory Seminar, The Ohio State University – March 2025
  Number Theory Seminar, Max Planck Institute for Mathematics – August 2025
  AMS Sectional Special Session on Arithmetic of Function Fields, Boston, MA – March 2026

• A volcanic approach to torsion of CM elliptic curves
  Oberseminar, Max Planck Institute for Mathematics – August 2025
  Lafayette College – February 2026
  Virginia Tech – February 2026 (slides) (slides 16:9)

• Shimura curve Atkin–Lehner quotients of genus at most two
  JMM AMS Special Session on Rethinking Number Theory, Washington, D.C. – January 2026

• Genus one Atkin–Lehner quotients of Shimura curves
  AMS Sectional Special Session on Recent Advances in the Mathematics of Cryptography – October 2025

• Shimura curves, gonalities, and low-degree points
  JMM AMS Special Session on Mathematics Informed by Computing, Seattle, WA – January 2025

• Low degree points on product surfaces (slides)
  MathFest, Indianapolis, IN – August 2024

• Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level (slides)
  ANTS XVI, MIT – July 2024
  Algebra and Number Theory Seminar, Dartmouth College – March 2024

• Heegner points on \(X_0(N)\) (slides) (video)
  Bhaskaracharya celebrating Harish-Chandra Lecture Series, Virtual – June 2024

• A volcanic approach to CM points on Shimura curves (slides)
  Algebra & Number Theory Seminar, Emory University – March 2023
  JMM AMS Special Session on Arithmetic Geometry Informed by Computation, Boston, MA – January 2023

• Classically integral quadratic forms excepting at most two values (slides)
  with M. Barowsky, W. Damron, A. Mejia, and N. Schock
  Joint Math Meetings, Atlanta, GA – January 2017

Other Conferences and Workshops Attended

• MPS Conference on The Eisenstein Ideal and Galois Representations
  Simons Foundation – May 2026

• The Mordell conjecture 100 years later
  MIT – July 2024

• Arizona Winter School - Abelian Varieties
  University of Arizona – March 2024

• Shimura curves in the LMFDB
  Dartmouth College – February 2024

• Collaborate@ICERM – Explicit Arithmetic of Shimura Curves
  ICERM – August 2023

• AMS Math Research Community – Explcit Computations with Stacks
  Beaver Hollow, New York – June 2023

• Integers 2023
  UGA – May 2023

• Georgia Algebraic Geometry Symposium (GAGS) 2023
  UGA – April 2023

• Virtual Number Theory and Arithmetic Geometry (VaNTAGe)
  Virtual – Fall 2020, Summer 2021, Fall 2021

• PCMI Graduate Summer School on
  Number Theory Informed by Computation
  Virtual – July 2021

• Mathematics Teacher-Scholar Symposium (MaTSS)
  Virtual – May 2021

• Connecticut Number Theory Conference (CTNT)
  Virtual – June 2020

• Front Range Number Theory Day
  Virtual – April 2020

• Arithmetic Geometry is Online In Zoom, Everyone (AGONIZE)
  Virtual – March 2020

• Arizona Winter School - Nonabelian Chabauty
  University of Arizona – March 2020

• Modular Forms, Arithmetic, and Women in Mathematics (MAAIM)
  Emory University – November 2019

• Midwest Arithmetic Geometry and Number Theory Series (MAGNTS)
  The Ohio State University – October 2019

• CMI-HIMR Summer School in Computational Number Theory
  University of Bristol – June 2019

• Connecticut Summer School in Number Theory and
  Conference on Elliptic Curves and Modular Forms (CTNT)
  University of Connecticut – May 2018